How useful is Graham's number?

Chris Smith asked Tony Padilla...

Tony - Graham's number, for a time, was the largest number ever to have appeared in a mathematical proof. It's named after Ron Graham, who is an American mathematician. He was solving some mathematical problems, these kinds of problems that mathematicians like to solve. It was very abstract. It's in some branch of mathematics called Ramsey theory. It involved hyper cubes in extra dimensions and all sorts of wonderful things. And he was trying to find bound for the results. Is this thing I'm trying to find, is it less than some number? And he was able to show it was less than this number, Graham's number, but this Graham's number is just ridiculously big. You take a number like a googol for example, which is a one with a hundred zeros. Well, Graham's number dwarfs that. Or take a googolplex, which is a one with a googol zeros, well, Graham's number dwarfs that. It's just off the scale, enormous. You can't write it down, you would run out of space in the universe to even try and write it down. So, to express it, we use a special type of notation that you have to invent to describe this thing. It's called Knuth's arrows, and it's like an extension of powers that you learned at school. Three squared, three cubed, that kind of thing. It's an extension of that that allows you to very rapidly get to very, very, very big numbers.

Tony - One of the things that I discussed about it in the videos and in the book was, I was trying to think about how big it is. And how can I express that? Well, what if you tried to imagine this number in your head, if you actually tried to picture its decimal expansion written out. Well, if you try to do that, Chris, then the inevitable outcome is that your head would collapse into a black hole. There's just too much information that, long before that happened, your head, it'd probably blow up, but if you managed to not blow up, it would collapse into a black hole.

Chris - So how did that guy's head not collapse into a black hole then?

Tony - Well, he used this clever notation, which allowed him to compact it. He wasn't thinking of it as a decimal expansion, right? Let's take the number 27. I could write down 27 or I could write down three cubed as a different way to express that number. So it's like that but in a much more enhanced way. So he had a way of expressing it that wasn't the big massive decimal expansion and so his head didn't collapse into a black hole.

Chris - But how did it change the world, the fact that he proved this?

Tony - It got in the Guinness Book of Records!

Chris - That really matters, okay.

Tony - I think what was important about it was that it showed what was provable. Proof theory for mathematicians is a really important thing. Understanding what's provable, what isn't provable, can you show that this can actually be answered, these are important questions in logic.